引入：绝对值
distance(:|a-b|)
properties(:(1)|x| geq 0),for all (x in R),and ("=” Leftrightarrow x=0)
((2):|a-b|=|b-a|(|x|=|-x|))
((3):|x+y| leq |x|+|y|),for all (x,y in R)
((|a-c| leq |a-b|+|b-c|))
度量空间
Distance function/metric space
Let (X) be a set.
(underline{Def:})A function (X imes X stackrel{d}{longrightarrow}mathbb{R})is called a distance function on (X)
1.(forall x,yin X),(d(x,y)geq 0) and ("=” Leftrightarrow x=y)
2.(forall x,yin X),(d(x,y)=d(y,x))
3.(forall x,y,z in X),(d(x,z)leq d(x,y)+d(y,z))
Example:
(mathfrak{A}:)
1.(x=(x_1,x_2,dots,x_m),y=(y_1,y_2,dots,y_m)in mathbb{R}^n)
(d_2(x,y):=sqrt{|x_1-y_1|^2+cdots+|x_m-y_m|^2}=|x-y|)
(d_2) is a metric on (mathbb{R}^n)(Cauchy inequality)
2.(d_1(x,y):=|x_1-y_1|+|x_2-y_2|+cdots+|x_m-y_m|)
3.(d_{infty}(x,y)=max{|x_1-y_1|,dots,|x_m-y_m|})
(mathfrak{B}:)
X:a set.For (x,y in X),let $$d(x,y):=left{
egin{aligned}
1&if&xleq y
0&if&x =y
end{aligned}
ight.
[$d(x,y)Rightarrow$the discrete metric
## 开集，闭集
we may generalize the definitions about limits and convergence to metric space
$underline{Def}$ Let $(X,d)$ be a metric space,$a_n(n in mathbb{N})$be a seq in $mathrm{X}$.and $mathcal{L}$in X
$a_n(n in mathbb{N})$converges to $mathcal{L}$
(1)For $r geq 0$and $x_0 in X$,we let $B_r(x_0)={x in X|d(x,x_0)leq r}$(open ball)
(2).S is an open set(of$(X,d)$),if $forall x in S$,$exists r >0$
（$B_r(x_0)subset S$）open ball $Rightarrow$open set
EX:
$(X,d):$metric space.$x_0 in X,r geq 0$
Show that:(1)$B_r(x_0)$is open
(2)${x in X|d(x,x_0）> r}$is open
warning:A subset $S$ of a topological space $(X, mathcal{T})$ is said to be clopen if it is both open and closed in $(X, mathcal{T})$
Example. $quad$ Let $X={a, b, c, d, e, f}$ and
]
au_{1}={X, emptyset,{a},{c, d},{a, c, d},{b, c, d, e, f}}
[We can see:
(i) the set ${a}$ is both open and closed;
(ii) the set ${b, c}$ is neither open nor closed;
(iii) the set ${c, d}$ is open but not closed;
(iv) the set ${a, b, e, f}$ is closed but not open.
In a discrete space every set is both open and closed, while in an indiscrete space$(X, au),$ all subsets of $X$ except $X$ and $emptyset$ are neither open nor closed.]